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the conditions of its motion are that it shall continue in
These elements, therefore, would not be cut by this plane, though all the other elements would. And if these parallel elements were projected upon the cutting plane, their projections could not touch the curve, though they would continually approach it, as we follow the curve down from the vertex. These projections are called asymtotes of the curve. It is important to observe that the plane, in the case of the hyperbola, though it cuts all the right-line elements of the surface excepting two, it does not cut them on the same side of the apex of the cone. But these elements being produced, the plane will cut the remainder beyond the apex. The curve, therefore, consists of two branches : The straight line joining their vertices is called the transverse axis ; and a straight line perpendicular to the middle of this, and in the plane of the curve, is generally assumed as the conjugate axis of the hyperbola. (4). If the angle which the cutting plane makes with the axis of the cone, is equal to that which the side makes with the axis, the section is a parabola. The parabola has but one axis. These curves, on account of this method of obtaining them, are usually called the Comic Sections.
59. If a point be conceived to move in the surface of a cylinder or a cone, with this condition, that its path shall, in every part, make the same angle with the axis of the cone or cylinder, it will, in each case, describe a spiral curve, which is a curve of double curvature. The more important curves of double curvature are those produced by the intersection of curve surfaces.
Of Curve Surfaces.
60. The curve surfaces of which geometry takes cognizance, are generated by lines, according to certain mathematical laws which serve to determine their parts and the character of their curvature.
As a plane surface is considered as consisting of an infinite number of straight lines, so we may conceive a curve surface to be composed of an infinite number of curves ; and this surface will be perfectly determined, when we have these curves and the law which connects each one of them with the next. A curve line, therefore, which moves in space, or which changes, at the same time, its magnitude and position, according to some determinate law, generates a curve surface. This surface may be considered as composed of the successive positions and forms which the line takes during its motion.
61. The most simple of curve surfaces are those of the cone and the cylinder.
A cylindrical surface is generated by a straight line moving in space, always parallel to itself, and directed in its motion by a curve line. The straight line is here called the generatrix, and the curve line the directrix. If the directrix be a circular curve, the surface will be that of the cylinder considered in the Elements.
The cylindrical surface may also be generated by the motion of a curve line along a straight line, each point in the curve moving in a direction parallel to the straight line. In this case the curve line is called the generatrir, and the straight line the directria.
If the cylindrical surface be cut by a plane parallel to the right-line generatrix, the section will be a straight line coincident with one position of this generatrix. The successive positions of this right-line generatrix we call the right-line elements of the cylindrical surface.
If this surface be cut by parallel planes making any angle with the right-line elements, these sections will be all the same curve; and each of these is an element of the cylindrical surface.
62. If the motion of the right-line generatrix were determined by the conditions, that this line should pass constantly through the same point, and that it should move along a curve line, the product of this motion would be a comical surface. If the curve line were the circumference of a circle, the conical surface thus produced would be that of the cone considered in the Elements. But it is not essential to a cylindrical or conical surface, as the terms are used in Descriptive Geometry, that the curvilinear directrix should be of a nature to return into itself.
Cones and cylinders are distinguished by their bases; they are called circular, elliptical, hyperbolical, or para
bolical, according as the base is a circle, an ellipse, a hyperbola or a parabola.
The conical surface may also be conceived to be generated by the motion of the curve which was before taken for the directrix ; the conditions being, that all the points in the curve shall move in straight lines converging to a fixed point. this point will be the summit of the cone.
63. As the right-line generatrix is considered as infinite, it is manifest that the conical surface generated by its motion will consist of two parts perfectly symetrical, but in opposite directions from their common summit. These two parts (called, in French, nappes) are considered as constituting but one and the same surface.
If a conical surface be cut by a plane passing through the summit, the section will be in a right-line element of the surface ; if it be cut by planes parallel to each other, none of which pass through the summit, these several sections will be similar curves, and are considered as curvilinear elements of the surface.
64. If the straight-line generatrix, instead of moving according to the conditions given for the cylinder or the cone, were to move along two straight lines or a straight line and a curve line not in the same plane, or along two curves not in the same plane and so situated that the straight line shall neither pass constantly through the same point, nor have its successive positions parallel, the surface generated by its motion will be what is called a warped surface.
65. It will be at once seen that there may be an infinite variety in the character and curvature of surfaces which fall under the general division of warped surfaces. We shall give one or two examples.
(1.) Let the right-line generatrix have two right-line directrices ; suppose that one of them is perpendicular to a horizontal plane, and the other inclined to this plane : and suppose that a condition of the motion of the generatrix, is, that it shall be constontly parallel to the horizontal plane. It will be readily perceived what is the kind of surface generated by this motion of the straight
(2.) Let there be one right-line directrix in the vertical plane, and perpendicular to the horizontal plane ; and let the other directrix be a circular arc of 900, whose plane is perpendicular to each of the co-ordinate planes, whose centre is in their common intersection, and whose extremities are in these planes respectively; and suppose the right-line generatrix to be always parallel to the horizontal plane. The body embraced by the surface generated by this straight line, and the two co-ordinate planes, is what was called by Wallis, a conoidal wedge.
66. Warped surfaces, although generated by straight lines, differ from cylindrical and conical surfaces, in a very important particular. The latter are capable of being developed or unrolled upon a plane ; the former are not capable of such developement.
That any surface may be developed, it is necessary that any two contiguous positions of the right-line generatrix should be in the same plane.
67. To understand the developement of a cylindrical surface, suppose a plane applied to it ; it is manifest that the plane will touch the cylindrical surface along one of its right-line elements. Let the cylinder be rolled upon the plane, the right line elements of its surface will come successively in contact with the plane, and when the cylinder has rolled once round, every right line element has touched the plane ; and the portion of the plane thus brought into contact with the cylindrical surface, in one entire revolution, is equivalent to the cylindrical surface, and may be considered its developement. Or suppose a piece of plane paper rolled about a cylinder, so as just to cover it, to be unrolled into a plane; it will represent the developement of the cylindrical surface.
The developement of a conical surface may be illustrated in a similar manner.
68. A more common method of illustrating the developement of these surfaces, is to begin with the prism and pyramid. One of the lateral faces of the prism, for instance, is applied to the plane upon which the surface is to be developed. The body is then made to revolve about the edge common to this face and the next, till this second face comes into the same plane ; the body is then